The many formulae for the number of Latin rectangles

Electronic Journal of Combinatorics

Volumn 17, Issue 1, 2010, Pages 1-46

  Stones, Douglas S ^a  

^a MONASH UNIVERSITY   (Australia)

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EID: 77955700853     PISSN: None     EISSN: 10778926     Source Type: Journal    
DOI: 10.37236/487     Document Type: Article

References (178)

1. A. A. Albert, Quasigroups II, Trans. Amer. Math. Soc., 55 (1944), pp. 401-419.
2. N. Alon, On the number of subgraphs of prescribed type of graphs with a given number of edges, Israel J. Math., 38 (1981), pp. 116-130.
3. R. Alter, How many Latin squares are there?, Amer. Math. Monthly, 82 (1975), pp. 632-634.
4. V. L. Arlazarov, A. M. Baraev, J. U. Gol'Fand and I. A. Faradzev, Construction with the aid of a computer of all Latin squares of order 8, Algorithmic Investigations in Combinatorics, (1978), pp. 129-141. In Russian.
5. r,s), European J. Combin., 1 (1980), pp. 9-17.
6. R. A. Bailey and P. J. Cameron, Latin squares: Equivalents and equivalence, (2003). http://designtheory.org/library/encyc/latinsq/g/topics/lsee.pdf.
7. S. E. Bammel and J. Rothstein, The number of 9 × 9 Latin squares, Discrete Math., 11 (1975), pp. 93-95.
8. R. C. Bose and B. Manvel, Introduction to Combinatorial Theory, Wiley, 1984.
9. L. J. Brant and G. L. Mullen, A note on isomorphism classes of reduced Latin squares of order 7, Util. Math., 27 (1985), pp. 261-263.
10. L. M. Brègman, Certain properties of nonnegative matrices and their permanents, Soviet Math. Dokl., 14 (1973), pp. 945-949.
11. J. W. Brown, Enumeration of Latin Squares and Isomorphism Detection in Finite Planes, PhD thesis, University of Los Angeles, 1966.
12. J. W. Brown, Enumeration of Latin squares with application to order 8, J. Combin. Theory, 5 (1968), pp. 177-184.
13. R. H. Bruck, Simple quasigroups, Bull. Amer. Math. Soc., 50 (1944), pp. 769-781.
14. R. H. Bruck Some results in the theory of quasigroups, Trans. Amer. Math. Soc., 55 (1944), pp. 19-52.
15. R. H. Bruck, Finite nets. I. Numerical invariants, Canadian J. Math., 3 (1951), pp. 94-107.
16. R. H. Bruck, Finite nets. II. Uniqueness and imbedding, Pacific J. Math., 13 (1963), pp. 421-457.
17. B. F. Bryant and H. Schneider, Principal loop-isotopes of quasigroups, Canad. J. Math., 18 (1966), pp. 120-125.
18. D. Bryant, B. M. Maenhaut, and I. M. Wanless, A family of perfect factorizations of complete bipartite graphs, J. Combin. Theory Ser. A, 98 (2002), pp. 328-342.
19. D. Bryant, B. M. Maenhaut, and I. M. Wanless, New families of atomic Latin squares and perfect 1-factorisations, J. Combin. Theory Ser. A, 113 (2004), pp. 608-624.
20. L. Carlitz, Congruences connected with three-line Latin rectangles, Proc. Amer. Math. Soc., 4 (1953), pp. 9-11.
21. A. Cayley, On Latin squares, Messenger of Math., 19 (1890), pp. 135-137. See [22], vol. 13, no. 903, pp. 55-57.
22. A. Cayley, The Collected Mathematical Papers of Arthur Cayley, Cambridge University Press, 1897.
23. G.-S. Cheon and I. M. Wanless, An update on Minc's survey of open problems involving permanents, Linear Algebra Appl., 403 (2005), pp. 314-342.
24. C. J. Colbourn, The complexity of completing partial Latin squares, Discrete Appl. Math., 8 (1984), pp. 25-30.
25. C. J. Colbourn, J. H. Dinitz, et al., The CRC Handbook of Combinatorial Designs, CRC Press, 1996.
26. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, 1974.
27. P. CsörgoO, A. Drápal and M. K. Kinyon, Buchsteiner loops, (2007). arXiv:0708.2358v1 [math.GR].
28. A. de Gennaro, How many Latin rectangles are there?, (2007). arXiv:0711.0527v1 [math.CO].
29. J. Dénes and A. D. Keedwell Latin Squares and their Applications, Academic Press, 1974.
30. J. Dénes and A. D. Keedwell,Latin squares and 1-factorizations of complete graphs. I. Connections between the enumeration of Latin squares and rectangles and r-factorizations of labeled graphs, Ars Combin., 25 (1988), pp. 109-126.
31. n using MacMahon's double partition idea, Util. Math., 34 (1988), pp. 73-83.
32. J. Dénes and A. D. Keedwell, Latin Squares: New Developments in the Theory and Applications, North- Holland, 1991.
33. J. Dénes and G. L. Mullen Enumeration formulas for Latin and frequency squares, Discrete Math., 111 (1993), pp. 157-163.
34. J. H. Dinitz and D. R. Stinson, Contemporary Design Theory: A Collection of Surveys, Wiley, 1992.
35. P. G. Doyle, The number of Latin rectangles, (2007). arXiv:math/0703896v1 [math.CO].
36. A. A. Drisko, On the number of even and odd Latin squares of order p + 1, Adv. Math., 128 (1997), pp. 20-35.
37. rp, Electron. J. Combin., 5 (1998). R28, 5 pp.
38. L. Duan, An upper bound for the number of normalized Latin square, Chinese Quart. J. Math., 21 (2006), pp. 585-589.
39. G. P. Egorichev, The solution of the van der Waerden problem for permanents, Adv. Math., 42 (1981), pp. 299-305.
40. P. Erdos and I. Kaplansky The asymptotic number of Latin rectangles, Amer. J. Math., 68 (1946), pp. 230-236.
41. M. J. Erickson, Introduction to Combinatorics, Wiley, 1996.
42. I. M. H. Etherington, Transposed algebras, Proc. Edinburgh Math. Soc. (2), 7 (1945), pp. 104-121.
43. L. Euler, Recherches sur une nouvelle espéce de quarrés magiques, Verh. Zeeuwsch. Gennot. Weten. Vliss., 9 (1782), pp. 85-239. Enestr̈om E530, Opera Omnia OI7, pp. 291-392.
44. R. M. Falcón Cycle structures of autotopisms of the Latin squares of order up to 11, Ars Combin., (2009). To appear.
45. R. M. Falcón J. Martín-Morales Gröbner bases and the number of Latin squares related to autotopisms of order ≤ 7, J. Symbolic Comput., 42 (2007), pp. 1142-1154.
46. D. I. Falikman, A proof of van der Waerden's conjecture on the permanent of a doubly stochastic matrix, Math. Notes, 29 (1981), pp. 475-479.
47. R. A. Fisher and F. Yates, The 6 × 6 Latin squares, Proc. Cambridge Philos. Soc., 30 (1934), pp. 492-507.
48. M. Frolov, Sur les permutations carrés, J. de Math. spéc., 4 (1890), pp. 8-11 and 25-30.
49. Z.-L. Fu, The number of Latin rectangles, Math. Practice Theory, (1992), pp. 40-41. In Chinese.
50. R. M. F. Ganfornina, Latin squares associated to principal autotopisms of long cycles. Application in cryptography, in Proceedings of Transgressive Computing, 2006, pp. 213-230.
51. The GAP Group, GAP - Groups, algorithms, programming - A system for computational discrete algebra. http://www.gap-system.org/.
52. P. Garcia, Major Percy Alexander MacMahon, PhD thesis, Open University, 2006.
53. I. M. Gessel, Counting three-line Latin rectangles, in Proceedings of the Colloque de Combinatoire Énumérative, Springer, 1986.
54. I. M. Gessel, Counting Latin rectangles, Bull. Amer. Math. Soc., 16 (1987), pp. 79-83.
55. S. G. Ghurye, A characteristic of species of 7 × 7 Latin squares, Ann. Eugenics, 14 (1948), p. 133.
56. D. G. Glynn, The conjectures of Alon-Tarsi and Rota in dimension prime minus one, SIAM J. Discrete Math., 24 (2010), pp. 394-399.
57. C. D. Godsil and B. D. McKay, Asymptotic enumeration of Latin rectangles, Bull. Amer. Math. Soc. (N.S.), 10 (1984), pp. 91-92.
58. C. D. Godsil and B. D. McKay, Asymptotic enumeration of Latin rectangles, J. Combin. Theory Ser. B, 48 (1990), pp. 19-44.
59. I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, 1983.
60. T. Granlund et al., GMP - GNU multiple precision arithmetic library. http://gmplib.org/
61. T. A. Green, Asymptotic Enumeration of Latin Rectangles and Associated Waiting Times, PhD thesis, Stanford University, 1984.
62. T. A. Green, Asymptotic enumeration of generalized Latin rectangles, J. Combin. Theory Ser. A, 51 (1989), pp. 149-160.
63. T. A. Green Erratum: "Asymptotic enumeration of generalized Latin rectangles", J. Combin. Theory Ser. A, 52 (1989), p. 322.
64. L. Habsieger and J. C. M. Janssen, The difference in even and odd Latin rectangles for small cases, Ann. Sci. Math. Québec, 19 (1995), pp. 69-77.
65. M. Hall, Jr., Distinct representatives of subsets, Bull. Amer. Math. Soc., 54 (1948), pp. 922-926.
66. C. Hämäläinen and N. J. Cavenagh, Transitive Latin bitrades, (2008). arXiv:0804.4663v1 [math.CO].
67. J. R. Hamilton and G. L. Mullen, How many i - j reduced Latin rectangles are there?, Amer. Math. Monthly, 87 (1980), pp. 392-394.
68. A. Heinze and M. Klin, Links between Latin squares, nets, graphs and groups: Work inspired by a paper of A. Barlotti and K. Strambach, Electron. Notes Discrete Math., 23 (2005), pp. 13-21.
69. A. Hulpke, P. Kaski and P. R. J. Östergård, The number of Latin squares of order 11, Math. Comp. To appear.
70. S. M. Jacob, The enumeration of the Latin rectangle of depth three by means of a formula of reduction, with other theorems relating to non-clashing substitutions and Latin squares, Proc. London Math. Soc. (2), 31 (1930), pp. 329-354.
71. A.-A. A. Jucys, The number of distinct Latin squares as a group-theoretical constant, J. Combin. Theory Ser. A, 20 (1976), pp. 265-272.
72. D. Jungnickel, Latin squares, their geometries and their groups. A survey, IMA Vol. Math. Appl., 21 (1990), pp. 166-225.
73. W. B. Jurkat and H. J. Ryser, Matrix factorizations of determinants and permanents, J. Algebra, 3 (1966), pp. 1-27.
74. V. Kala and A. D. Keedwell, Addendum to "The existence of Buchsteiner and conjugacy-closed quasigroups", European J. Combin., 30 (2009), p. 1386.
75. A. D. Keedwell, The existence of Buchsteiner and conjugacy-closed quasigroups, European J. Combin., 30 (2009), pp. 1382-1385.
76. S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), pp. 119-127.
77. B. Kerby and J. D. H. Smith, Quasigroup automorphisms and symmetric group characters, Comment. Math. Univ. Carol., 51 (2010), pp. 279-286.
78. B. Kerby and J. D. H. Smith, Quasigroup automorphisms and the Norton-Stein complex, Proc. Amer. Math. Soc. To appear.
79. M. K. Kinyon, K. Kunen and J. D. Phillips, Diassociativity in conjugacy closed loops, Comm. Algebra, 32 (2004), pp. 767-786.
80. D. Klyve and L. Stemkoski, Graeco-Latin squares and a mistaken conjecture of Euler, Cambridge Mathematical Journal, 37 (2006), pp. 2-15.
81. G. Kolesova, C. W. H. Lam and L. Thiel, On the number of 8 × 8 Latin squares, J. Combin. Theory Ser. A, 54 (1990), pp. 143-148.
82. N. Y. Kuznetsov, Estimating the number of Latin rectangles by the fast simulation method, Cybernet. Systems Anal., 45 (2009), pp. 69-75.
83. C. Laywine and G. L. Mullen, Discrete Mathematics using Latin Squares, Wiley, 1998.
84. H. Liang and F. Bai, An upper bound for the permanent of (0, 1)-matrices, Linear Algebra Appl., 377 (2004), pp. 291-295.
85. F. W. Light, Jr., A procedure for the enumeration of 4 × n Latin rectangles, Fibonacci Quart., 11 (1973), pp. 241-246.
86. F. W. Light, Jr, Enumeration of truncated Latin rectangles, Fibonacci Quart., 17 (1979), pp. 34-36.
87. P. A. MacMahon, A new method in combinatory analysis, with applications to Latin squares and associated questions, Trans. Camb. Phil. Soc., 16 (1898), pp. 262-290.
88. P. A. MacMahon, Combinatorial analysis. the foundations of a new theory, Philos. Trans. Roy. Soc. London Ser. A, 194 (1900), pp. 361-386.
89. P. A. MacMahon, Combinatory Analysis, Chelsea, 1960.
90. P. A. MacMahon, Percy Alexander MacMahon: collected papers/edited by George E. Andrews, MIT Press, 1986.
91. B. M. Maenhaut and I. M. Wanless, Atomic Latin squares of order eleven, J. Combin. Des., 12 (2004), pp. 12-34.
92. M. Marcus, Henryk Minc - a biography, Linear Multilinear Algebra, 51 (2003), pp. 1-10.
93. B. D. McKay, nauty - Graph isomorphism. http://cs.anu.edu.au/~bdm/nauty/
94. B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups, and loops, J. Combin. Des., 15 (2007), pp. 98-119.
95. B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combin., 2 (1995). N3, 4 pp.
96. B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Comb., 9 (2005), pp. 335-344.
97. H. Minc, Upper bounds for permanents of (0, 1)-matrices, Bull. Amer. Math. Soc., 69 (1963), pp. 789-791.
98. H. Minc, An inequality for permanents of (0, 1)-matrices, J. Combinatorial Theory, 2 (1967), pp. 321-326.
99. H. Minc, On lower bounds for permanents of (0, 1) matrices, Proc. Amer. Math. Soc., 22 (1969), pp. 117-123.
100. H. Minc, Permanents, Addison-Wesley, 1978.
101. H. Minc, Theory of permanents, 1978-1981, Linear Multilinear Algebra, 12 (1982/83), pp. 227-263.
102. H. Minc, Theory of permanents, 1982-1985, Linear Multilinear Algebra, 21 (1987), pp. 109-148.
103. D. S. Mitrinović, R. R. Janić and J. D. Kečkić Albert Sade (1898-1973), Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 412-460 (1973), pp. 243-246.
104. W. O. J. Moser, The number of very reduced 4 × n Latin rectangles, Canad. J. Math., 19 (1967), pp. 1011-1017.
105. G. L. Mullen, How many i - j reduced Latin squares are there?, Amer. Math. Monthly, 82 (1978), pp. 751-752.
106. G. L. Mullen and C. Mummert, Finite Fields and Applications, American Mathematical Society, 2007.
107. G. L. Mullen and D. Purdy, Some data concerning the number of Latin rectangles, J. Combin. Math. Combin. Comput., 13 (1993), pp. 161-165.
108. G. P. Nagy and P. Vojtěchovský, LOOPS - Computing with quasigroups and loops in GAP. http://www.math.du.edu/loops/
109. G. P. Nagy and P. Vojtěchovský, Computing with small quasigroups and loops, Quasigroups Related Systems, 15 (2007), pp. 77-94.
110. J. R. Nechvatal, Counting Latin Rectangles, PhD thesis, University of Southern California, 1979.
111. J. R. Nechvatal, Asymptotic enumeration of generalized Latin rectangles, Util. Math., 20 (1981), pp. 273-292.
112. D. A. Norton and S. K. Stein, An integer associated with Latin squares, Proc. Amer. Math. Soc., 7 (1956), pp. 331-334.
113. D. A. Norton and S. K. Stein, Cycles in algebraic systems, Proc. Amer. Math. Soc., 7 (1956), pp. 999-1004.
114. H. W. Norton, The 7 × 7 squares, Ann. Eugenics, 2 (1939), pp. 269-307.
115. P. E. O'Neil, Asymptotics and random matrices with row-sum and column-sum restrictions, Bull. Amer. Math. Soc., 75 (1969), pp. 1276-1282.
116. K. T. Phelps, Latin square graphs and their automorphism groups, Ars Combin., 7 (1979), pp. 273-299.
117. K. T. Phelps, Automorphism-free Latin square graphs, Discrete Math., 31 (1980), pp. 193-200.
118. C. R. Pranesachar, Enumeration of Latin rectangles via SDR's, in Lecture Notes in Math., Springer, 1981, pp. 380-390.
119. D. A. Preece, Classifying Youden rectangles, J. Roy. Statist. Soc. Ser. B, 28 (1966), pp. 118-130.
120. J. Riordan, Three-line Latin rectangles, Amer. Math. Monthly, 51 (1944), pp. 450-452.
121. J. Riordan, A recurrence relation for three-line Latin rectangles, Amer. Math. Monthly, 59 (1952), pp. 159-162.
122. J. Riordan, Book review: Percy Alexander MacMahon: Collected Papers, Volume I, Combinatorics, Bull. Amer. Math. Soc. (N.S.), 2 (1980), pp. 239-241.
123. G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrsch. und Verw. Gebiete, 2 (1964), pp. 340-368.
124. H. J. Ryser, Combinatorial Mathematics, Springer, 1963.
125. e ordre, conjecture pour les ordres supérieurs, privately published, (1948). 8 pp.
126. A. A. Sade, Omission dans les listes de Norton pour les carr ́es 7 × 7, J. Reine Angew. Math., 189 (1951), pp. 190-191.
127. A. A. Sade, An omission in Norton's list of 7 × 7 squares, Ann. Math. Stat., 22 (1951), pp. 306-307.
128. A. A. Sade, Quasigroupes parastrophiques. Expressions et identités, Math. Nachr., 20 (1958), pp. 73-106.
129. A. A. Sade, Quasigroupes isotopes. Autotopies d'un groupe, Annales de la Société Scientifique de Bruxelles, 81 (1967), pp. 231-239.
130. A. A. Sade, Autotopies des quasigroupes et des systémes associatifs, Arch. Math. (Brno), 4 (1968), pp. 1-23.
131. A. A. Sade, Morphismes de quasigroupes. Tables, Univ. Lisboa Revista Fac. Ci. A, 13 (1970/71), pp. 149-172.
132. P. N. Saxena, A simplified method of enumerating Latin squares by MacMahon's differential operators; I. the 6 × 6 squares, J. Indian Soc. Agricultural Statist., 2 (1950), pp. 161-188.
133. P. N. Saxena, A simplified method of enumerating Latin squares by MacMahon's differential operators; II. the 7×7 squares, J. Indian Soc. Agricultural Statist., 3 (1951), pp. 24-79.
134. E. Schönhardt, Über lateinische Quadrate und Unionen, J. Reine Angew. Math., 163 (1930), pp. 183-229.
135. A. Schrijver, A short proof of Minc's conjecture, J. Combin. Theory Ser. A, 25 (1978), pp. 80-83.
136. E. Schröder, Ueber v. Staudt's Rechnung mit Würfen und verwandte Processe, Math. Ann., 10 (1876), pp. 289-317.
137. E. Schröder, Ueber eine eigenthümliche Bestimmung einer Function durch formale Anforderungen, J. Reine Angew. Math., 90 (1881), pp. 189-220.
138. J.-Y. Shao and W.-D. Wei, A formula for the number of Latin squares, Discrete Math., 110 (1992), pp. 293-296.
139. J. B. Shaw, On parastrophic algebras, Trans. Amer. Math. Soc., 16 (1915), pp. 361-370.
140. R. Sinkhorn, Concerning a conjecture of Marshall Hall, Proc. Amer. Math. Soc., 21 (1969), pp. 197-201.
141. I. Skau, A note on the asymptotic number of Latin rectangles, European J. Combin., 19 (1998), pp. 617-620.
142. N. J. A. Sloane, The on-line encyclopedia of integer sequences. Published electronically at http://www.research.att.com/~njas/sequences/
143. B. Smetaniuk, A new construction of Latin squares. II. The number of Latin squares is strictly increasing, Ars Combin., 14 (1982), pp. 131-145.
144. J. D. H. Smith, Homotopies of central quasigroups. Submitted.
145. R. P. Stanley, Differentiably finite power series, European J. Combin., 1 (1980), pp. 175-188.
146. C. M. Stein, Asymptotic evaluation of the number of Latin rectangles, J. Combin. Theory Ser. A, 25 (1978), pp. 38-49.
147. C. M. Stein, Approximate Computation of Expectations, Institute of Mathematical Statistics, 1986.
148. C. M. Stein, A way of using auxiliary randomization, in Probability Theory, Walter de Gruyter & Co., 1992, pp. 159-180.
149. S. K. Stein, Foundations of quasigroups, Proc. Natl. Acad. Sci. USA, 42 (1956), pp. 545-546.
150. S. K. Stein, On the foundations of quasigroups, Trans. Amer. Math. Soc., 85 (1957), pp. 228-256.
151. D. S. Stones, The parity of the number of quasigroups. Submitted.
152. D. S. Stones, (2009). http://code.google.com/p/latinrectangles/downloads/list.
153. D. S. Stones, On the Number of Latin Rectangles, PhD thesis, Monash University, 2010. http://arrow.monash.edu.au/hdl/1959.1/167114.
154. D. S. Stones and I. M. Wanless, Compound orthomorphisms of the cyclic group, Finite Fields Appl., 16 (2010), pp. 277-289.
155. D. S. Stones, A congruence connecting Latin rectangles and partial orthomorphisms. Submitted.
156. D. S. Stones, How not to prove the Alon-Tarsi Conjecture. Submitted.
157. D. S. Stones, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A, 117 (2010), pp. 204-215.
158. G. Tarry, Le probléme des 36 officers, Ass. Franc. Paris, 29 (1901), pp. 170-203.
159. H. A. Thurston, Certain congruences on quasigroups, Proc. Amer. Math. Soc., 3 (1952), pp. 10-12.
160. A. N. Timashov, On permanents of random doubly stochastic matrices and on asymptotic estimates for the number of Latin rectangles and Latin squares, Discrete Math. Appl., 12 (2002), pp. 431-452.
161. B. L. van der Waerden, Aufgabe 45, J. de Math. spéc., 35 (1926), p. 117.
162. D. C. van Leijenhorst, Symmetric functions, Latin squares and Van der Corput's "scriptum 3", Expo. Math., 18 (2000), pp. 343-356.
163. J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992.
164. I. M. Wanless, Perfect factorisations of bipartite graphs and Latin squares without proper subrectangles, Electron. J. Combin., 6 (1999). R9, 16 pp.
165. n, Linear Algebra Appl., 373 (2003), pp. 153-167.
166. I. M. Wanless, Atomic Latin squares based on cyclotomic orthomorphisms, Electron. J. Combin., 12 (2005). R22, 23 pp.
167. I. M. Wanless and E. C. Ihrig, Symmetries that Latin squares inherit from 1-factorizations, J. Combin. Des., 13 (2005), pp. 157-172.
168. M. B. Wells, The number of Latin squares of order eight, J. Combin. Theory, 3 (1967), pp. 98-99.
169. M. B. Wells, Elements of Combinatorial Computing, Pergamon Press, 1971.
170. H. Whitney, A logical expansion in mathematics, Bull. Amer. Math. Soc., 38 (1932), pp. 572-579.
171. H. S. Wilf, What is an answer?, Amer. Math. Monthly, 89 (1982), pp. 289-292.
172. K. Yamamoto, On the asymptotic number of Latin rectangles, Japan. J. Math., 21 (1951), pp. 113-119.
173. K. Yamamoto, Note on enumeration of 7 × 7 Latin squares, Bull. Math. Statist., 5 (1952), pp. 1-8.
174. K. Yamamoto, On Latin squares, Sǔgaku, 12 (1960/61), pp. 67-79. In Japanese.
175. K. Yamamoto, On the number of Latin rectangles, Res. Rep. Sci. Div. Tokyo Womens' Univ., 19 (1969), pp. 86-97.
176. P. Zappa, The Cayley determinant of the determinant tensor and the Alon-Tarsi Conjecture, Adv. in Appl. Math., 13 (1997), pp. 31-44.
177. D. Zeilberger, Enumerative and algebraic combinatorics, in Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 550-561.
178. C. Zhang and J. Ma, Counting solutions for the N-queens and Latin square problems by Monte Carlo simulations, Phys. Rev. E, 79 (2009). 016703.